by vinay karwasraLast Modified: March 28, 2018

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Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions MCOM sem 2 Delhi University:- we will provide complete details of Cost behavior analysis of Queuing models for Quantitative Techniques For Business Decisions MCOM sem 2 Delhi University in this article.

Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange.The ideas have since seen applications including telecommunication, traffic engineering, computing and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management.

Single queueing nodes are usually described using Kendall’s notation in the form A/S/C where A describes the time between arrivals to the queue, S the size of jobs and C the number of servers at the node. Many theorems in queueing theory can be proved by reducing queues to mathematical systems known as Markov chains, first described by Andrey Markov in his 1906 paper.

Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory in 1909. He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920. In Kendall’s notation:

M stands for Markov or memoryless and means arrivals occur according to a Poisson process

D stands for deterministic and means jobs arriving at the queue require a fixed amount of service

k describes the number of servers at the queueing node (k = 1, 2,…). If there are more jobs at the node than there are servers then jobs will queue and wait for service

The M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process and have exponentially distributed service requirements. In an M/G/1 queue the G stands for general and indicates an arbitrary probability distribution. The M/G/1 model was solved by Felix Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek–Khinchine formula.

After the 1940s queueing theory became an area of research interest to mathematicians. In 1953 David George Kendall solved the GI/M/k queue and introduced the modern notation for queues, now known as Kendall’s notation. In 1957 Pollaczek studied the GI/G/1 using an integral equation.John Kingman gave a formula for the mean waiting time in a G/G/1 queue: Kingman’s formula.

The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered.

Problems such as performance metrics for the M/G/k queue remain an open problem.

At least one customer is assumed to always be present, so the server is never idle, e.g., sufficient raw material for a machine.

Customer is pending when the customer is outside the queueing system, e.g., machine-repair problem: a machine is “pending” when it is operating, it becomes “not pending” the instant it demands service form the repairman.

Runtime of a customer is the length of time from departure from the queueing system until that customer’s next arrival to the queue, e.g., machine-repair problem, machines are customers and a runtime is time to failure.

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